Semirings of continuous nonnegative functions: divisibility, ideals, congruences

Authors investigate the properties of divisibility (GCD, LCM, to be Bezout semiring) in semirings of continuous nonnegative real-valued functions on a topological space $X$. The correspondences between the lattice of ideals of the ring $C(X)$ and the lattice of ideals of the semiring $C^{+}(X)$ are considered. New characterizations of $F$-spaces are obtained. Congruences on abstract semirings are studied. Maximal congruences of semirings $C^+(X)$ are described. It is shown that all congruences on a semifield $U(X)$ of all continuous pozitive functions on $X$ are ideal congruences if and only if $X$ is the pseudocompact space.